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Abstract

We have performed magnetotransport measurements on a multi-layer graphene flake. At
the crossing magnetic field Bc, an approximately temperature-independent point in the measured longitudinal resistivity
ρxx, which is ascribed to the direct insulator-quantum Hall (I-QH) transition, is observed.
By analyzing the amplitudes of the magnetoresistivity oscillations, we are able to
measure the quantum mobility μq of our device. It is found that at the direct I-QH transition, μqBc ≈ 0.37 which is considerably smaller than 1. In contrast, at Bc, ρxx is close to the Hall resistivity ρxy, i.e., the classical mobility μBc is ≈ 1. Therefore, our results suggest that different mobilities need to be introduced
for the direct I-QH transition observed in multi-layered graphene. Combined with existing
experimental results obtained in various material systems, our data obtained on graphene
suggest that the direct I-QH transition is a universal effect in 2D.

Keywords:

Background

Graphene, which is an ideal two-dimensional system [1], has attracted a great deal of worldwide interest. Interesting effects such as Berry's
phase [2,3] and fractional quantum Hall effect [4-6] have been observed in mechanically exfoliated graphene flakes [1]. In addition to its extraordinary electrical properties, graphene possesses great
mechanical [7], optical [8], and thermal [9] characteristics.

The insulator-quantum Hall (I-QH) transition [10-13] is a fascinating physical phenomenon in the field of two-dimensional (2D) physics.
In particular, a direct transition from an insulator to a high Landau-level filling
factor ν > 2 QH state which is normally dubbed as the direct I-QH transition continues to
attract interest [14]. The direct I-QH transition has been observed in various systems such as SiGe hole
gas [14], GaAs multiple quantum well devices [15], GaAs two-dimensional electron gases (2DEGs) containing InAs quantum dots [16-18], a delta-doped GaAs quantum well with additional modulation doping [19,20], GaN-based 2DEGs grown on sapphire [21] and on Si [22], InAs-based 2DEGs [23], and even some conventional GaAs-based 2DEGs [24], suggesting that it is a universal effect. Although some quantum phase transitions,
such as plateau-plateau transitions [25] and metal-to-insulator transitions [26-29], have been observed in single-layer graphene and insulating behavior has been observed
in disordered graphene such as hydrogenated graphene [30-33], graphene exposed to ozone [34], reduced graphene oxide [35], and fluorinated graphene [36,37], the direct I-QH transition has not been observed in a graphene-based system. It
is worth mentioning that the Anderson localization effect, an important signature
of strong localization which may be affected by a magnetic field applied perpendicular
to the graphene plane, was observed in a double-layer graphene heterostructure [38], but not in single-layer pristine graphene. Moreover, the disorder of single graphene
is normally lower than those of multi-layer graphene devices. Since one needs sufficient
disorder in order to see the I-QH transition [11], multi-layer graphene seems to be a suitable choice for studying such a transition
in a pristine graphene-based system. Besides, the top and bottom layers may isolate
the environmental impurities [39-42], making multi-layer graphene a stable and suitable system for observing the I-QH
transition.

In this paper, we report magnetotransport measurements on a multi-layer graphene flake.
We observe an approximately temperature-independent point in the measured longitudinal
resistivity ρxx which can be ascribed to experimental evidence for the direct I-QH transition. At
the crossing field Bc in which ρxx is approximately T-independent, ρxx is close to ρxy. In contrast, the product of the quantum mobility determined from the oscillations
in ρxx and Bc is ≈ 0.37 which is considerably smaller than 1. Thus, our experimental results suggest
that different mobilities need to be introduced when considering the direct I-QH transition
in graphene-based devices.

Methods

A multi-layer graphene flake, mechanically exfoliated from natural graphite, was deposited
onto a 300-nm-thick SiO2/Si substrate. Optical microscopy was used to locate the graphene flakes, and the
thickness of multi-layer graphene is 3.5 nm, checked by atomic force microscopy. Therefore,
the layer number of our graphene device is around ten according to the 3.4 Å graphene
inter-layer distance [1,43]. Ti/Au contacts were deposited on the multi-layer graphene flake by electron-beam
lithography and lift-off process. The multi-layer graphene flake was made into a Hall
bar pattern with a length-to-width ratio of 2.5 by oxygen plasma etching process [44]. Similar to the work done using disordered graphene, our graphene flakes did not
undergo a post-exfoliation annealing treatment [45,46]. The magnetoresistivity of the graphene device was measured using standard AC lock-in
technique at 19 Hz with a constant current I = 20 nA in a He3 cryostat equipped with a superconducting magnet.

Results and discussion

Figure 1 shows the curves of longitudinal and Hall resistivity ρxx(B) and ρxy(B) at T = 0.28 K. Features of magnetoresistivity oscillations accompanied by quantum Hall
steps are observed at high fields. In order to further study these results, we analyze
the positions of the extrema of the magnetoresistivity oscillations in B as well as the heights of the QH steps. Although the steps in the converted Hall
conductivity ρxy are not well quantized in units of 4e2/h, they allow us to determine the Landau-level filling factor as indicated in the inset
of Figure 1. The carrier density of our device is calculated to be 9.4 × 1016 m−2 following the procedure described in [47,48].

Figure 1.Longitudinal and Hall resistivity ρxx(B) and ρxy(B) at T = 0.28 K. The inset shows the converted ρxy (in units of 4e2/h ) and ρxx as a function of B.

We now turn to our main experimental finding. Figure 2 shows the curves of ρxx(B) and ρxy(B) as a function of magnetic field at various temperatures T. An approximately T-independent point in the measured ρxx at Bc = 3.1 T is observed. In the vicinity of Bc, for B < Bc, the sample behaves as a weak insulator in the sense that ρxx decreases with increasing T. For B > Bc, ρxx increases with increasing T, characteristic of a quantum Hall state. At Bc, the corresponding Landau-level filling factor is about 125 which is much bigger
than 1. Therefore, we have observed evidence for a direct insulator-quantum Hall transition
in our multi-layer graphene. The crossing points for B > 5.43 T can be ascribed to approximately T-independent points near half filling factors in the conventional Shubnikov-de Haas
(SdH) model [17].

Figure 2.Longitudinal and Hall resistivity ρxx(B) and ρxy(B) at various temperatures T. An approximately T-independent point in ρxx is indicated by a crossing field Bc.

By analyzing the amplitudes of the observed SdH oscillations at various magnetic fields
and temperatures, we are able to determine the effective mass m* of our device which is an important physical quantity. The amplitudes of the SdH
oscillations ρxx is given by [49]:

(1)

where , ρ0, kB, h, and e are a constant, the Boltzmann constant, Plank's constant, and electron charge, respectively.
When , we have

(2)

where C1 is a constant. Figure 3 shows the amplitudes of the SdH oscillations at a fixed magnetic field of 5.437 T.
We can see that the experimental data can be well fitted to Equation 2. The measured
effective mass ranges from 0.06m0 to 0.07m0 where m0 is the rest mass of an electron. Interestingly, the measured effective mass is quite
close to that in GaAs (0.067m0).

Figure 3.Amplitudes of the observed oscillations Δρxxat B = 5.437 T at different temperatures. The curve corresponds to the best fit to Equation 2.

In our system, for the direct I-QH transition near the crossing field, ρxx is close to ρxy. In this case, the classical Drude mobility is approximately the inverse of the crossing
field 1/Bc. Therefore, the onset of Landau quantization is expected to take place near Bc[50]. However, it is noted that Landau quantization should be linked with the quantum
mobility, not the classical Drude mobility [19]. In order to further study the observed I-QH transition, we analyze the amplitudes
of the magnetoresistivity oscillations versus the inverse of B at various temperatures. As shown in Figure 4, there is a good linear fit to Equation 1 which allows us to estimate the quantum
mobility to be around 0.12 m2/V/s. Therefore, near μqBc≈ 0.37 which is considerably smaller than 1. Our results obtained on multi-layered
graphene are consistent with those obtained in GaAs-based weakly disordered systems
[19,21].

Figure 4.as a function of the inverse of the magnetic field 1/B. The solid line corresponds to the best fit to Equation 1.

It has been shown that the elementary neutral excitations in graphene in a high magnetic
field are different from those of a standard 2D system [51]. In this case, the particular Landau-level quantization in graphene yields linear
magnetoplasmon modes. Moreover, instability of magnetoplasmons can be observed in
layered graphene structures [52]. Therefore, in order to fully understand the observed I-QH transition in our multi-layer
graphene sample, magnetoplasmon modes as well as collective phenomena may need to
be considered. The spin effect should not be important in our system [53]. At present, it is unclear whether intra- and/or inter-graphene layer interactions
play an important role in our system. Nevertheless, the fact that the low-field Hall
resistivity is nominally T-independent suggests that Coulomb interactions do not seem to be dominant in our
system.

Conclusion

In conclusion, we have presented magnetoresistivity measurements on a multi-layered
graphene flake. An approximately temperature-independent point in ρxx is ascribed to the direct I-QH transition. Near the crossing field Bc, ρxx is close to ρxy, indicating that at Bc, the classical mobility is close to 1/Bc such that Bc is close to 1. On the other hand, μqBc≈ 0.37 which is much smaller than 1. Therefore, different mobilities must be considered
for the direct I-QH transition. Together with existing experimental results obtained
on various material systems, our new results obtained in a graphene-based system strongly
suggest that the direct I-QH transition is a universal effect in 2D.

Abbreviations

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

CC and LHL performed the experiments. CC, TO, and AMM fabricated the device. NA, YO,
and JPB coordinated the project. TPW and STL provided key interpretation of the data.
CC and CTL drafted the paper. All the authors read and agree the final version of
the paper.

Acknowledgments

This work was funded by the National Science Council (NSC), Taiwan (grant no: NSC
99-2911-I-002-126 and NSC 101-2811-M-002-096). CC gratefully acknowledges the financial
support from Interchange Association, Japan (IAJ) and the NSC, Taiwan for providing
a Japan/Taiwan Summer Program student grant.